US7249081B2 - Load aware optimization - Google Patents
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Abstract
Description
TABLE 1 |
Exemplary Set of Factors |
Group | Factor | ||
Cash: | Short Term US Bonds (core class) | ||
Bonds: | Intermediate-term US Bonds (core class) | ||
Long-term US Bonds (core class) | |||
US Corporate Bonds | |||
US Mortgage Backed Securities | |||
Non-US Government Bonds | |||
Equities: | Large Cap Stock-Value | ||
Large Cap Stock-Growth | |||
Mid Cap Stock-Value | |||
Mid Cap Stock-Growth | |||
Small Cap Stock-Value | |||
Small Cap Stock-Growth | |||
Foreign: | International Equity-Europe | ||
International Equity-Pacific | |||
International Equity-Emerging Markets | |||
Γt=[(1−γ)·(1+R t)]−1 (EQ #1)
The adjusted gross return is then equal to (1+Γt). In equation (1), the variable γ is called the adjustment, and given a fixed scenario of returns {R1, R2, . . . RT} during the holding period, it is assumed to be the same over all time intervals in the holding period. However, the adjustment will typically vary from one scenario to the next. An equation to determine the adjustment is derived by equating the ending wealth of the loaded financial product to the ending wealth of the equivalent no-load financial product. This equation can be solved approximately (either numerically or analytically) for the adjustment. The adjustment will depend on the load fee, the initial invested wealth, the contributions and withdrawals, the holding period, and the scenario of random returns of the financial product during the holding period. Therefore the adjustment is a random variable with respect to the set of random scenarios of returns on the loaded financial product.
E[Γ t ]=E[(1−γ)·R t ]−E[γ] (EQ #2)
An approximate value for E[Γt] can be computed numerically using Monte Carlo simulations.
cov└ΓA,t,ΓB,t┘=cov└(1−γA)·(1+R A,t),(1−γB)·(1+R B,t)┘ (EQ #3)
Here, the expectations underlying this covariance are with respect to both of the random processes that generate the stochastic returns of the loaded financial products. Approximate values for the covariance can be computed numerically using Monte Carlo techniques.
E[Γ t ]=E[R t ]−E[γ] (EQ #4)
Similarly, equation (3) for the covariance of adjusted returns reduces to:
cov└ΓA,t,ΓB,t┘=cov└(1+R A,t),(1+R B,t)┘=cov└R A,t ,R B,t┘ (EQ #5)
Hence, the covariance of adjusted returns is approximately equal to the covariance of unadjusted returns, and the latter can be used for the former. Further, it is apparent that the major impact of loads on mean-variance optimization is via the adjustment to the expected returns. These approximations are confirmed by tests using Monte Carlo techniques.
where ∥x∥=max(x,0), and G(t,T), the gross return of the financial product with loads over all the intervals from t to T, is given by:
It is observed from the left hand side of equation (6) that load fees are paid on only a positive beginning increment of wealth and on contributions, but not on withdrawals.
This approximation neglects quadratic and all higher powers of the load fee. Since practical load fees are much less than 10%, the magnitude of terms neglected is much less than 1%.
Again, these approximations are confirmed by tests using Monte Carlo techniques. It is seen from equation (9) that the expected value of the returns of the equivalent no-load financial product depends on the expected value of the returns for the financial product with front-end loads, the holding period, the initial wealth, and the future and current contributions and withdrawals, as well as the load fee.
W t =x o,i ·W p
ΔW i=(x i −x o,i)·W p
C i,t =x i ·C p,t (EQ #10)
Here, Wp represents the beginning wealth of the total portfolio, Wt is the beginning wealth of the ith financial product, and xo,i is the beginning fraction of portfolio wealth in the ith financial product. Similarly, xi is the target wealth fraction for the ith financial product. The sums of the beginning fractions and target fractions are both equal to one. The beginning wealth increment for the ith financial product is ΔWi. Since an initial contribution or withdrawal can be made at the beginning of the first period, the increment in wealth for the portfolio as a whole is taken to be zero, without loss of generality. Lastly, Ci,t is the contribution (or withdrawal) to the ith financial product at the beginning of the tth period. As an approximation, this quantity is taken to be the product of the target fraction and the contribution to the portfolio at a whole, which is denoted by Cp,t. This approximation neglects any incremental contributions and withdrawals that occur because of transfers of wealth between products during a periodic rebalancing.
x i ·E[Γ i,t]=(e i−δi)·x i −p i ·∥x i −x o,i∥ (EQ #11)
where the parameters δi and pi are given by:
The parameters δi and pi are referred to as the reduction parameter and penalty parameter, respectively. These parameters are functions of the load fee, initial portfolio wealth, future and current portfolio contributions and withdrawals, expected unadjusted returns, and the holding period. The product of the reduction parameter δi and the target mix xt represents a cost in expected return due to the ongoing payment of loads through contributions. The product of the penalty parameter p1 and any increment to the target mix above the initial mix ∥xi−xo,i∥ represents a cost in expected return due to an initial rebalancing of wealth from the initial mix to the target mix.
Φ(x)=e T x−(x T Cx)/τ
such that
iTx=1
1≦x≦u (EQ #13)
In equations (13), the superscript T denotes a vector transpose. The parameter τ is a specified risk tolerance parameter, x is a vector of target mixes, e is a vector of expected returns, i is a vector of ones, 1 is a vector of lower bounds, and u is a vector of upper bounds on the target mixes. All vectors are N dimensional column vectors, where N is the number of financial products considered for the portfolio. The ith element of each vector corresponds to the ith financial product. The (N×N) covariance matrix C has the covariance of returns between the ith and jth financial products as the element in its ith row and jth column. In the first equation, the objective function Φ(x) is to be maximized by computing an optimal target mix x* from the set of all feasible target mix vectors x. The objective function is a quadratic function of the target mix and a linear combination of the portfolio's expected return (eTx, linear in x) and the portfolio's variance of returns (xTCx, quadratic in x). The second equation, a feasibility constraint, requires that the sum of the elements of the target mix to be equal to one. This equation is referred to as the budget constraint. The bounds allow an investor to specify minimum and maximum amounts of a financial product he desires to hold in his portfolio. The investor can also specify an explicit amount to hold by setting the upper and lower bounds to the same explicit value. A typical value for a lower bound is zero; this is a no short-sell constraint. The upper and lower bounds are assumed to be feasible with respect to each other, and with respect to the budget constraint.
i j T ·x=ƒ j , jε{1, . . . ,M} (EQ #14)
In equation (14), the vector ij is an N dimensional column vector with ones for the elements that correspond to financial products in the jth account, and zeros elsewhere. The values of the account fractions fj are given and represent the amount of total portfolio wealth to be allocated to the jth account. The sum of the account fractions typically equals one. Note that when there is just a single account, equations (14) reduce to the previous single account formulation given in equations (13). The typical portfolio optimization problem with multiple accounts has the following properties. The objective function is a quadratic function of N decision variables (the elements of x). There are simple upper and lower bounds on the decision variables (the vectors l and u). There are exactly M equality (budget) constraints on the feasible set of decision variables.
Φ(x)=(e−δ)T x−p T ∥x−x o∥−(xCx)/τ (EQ #15)
In equation (15) the N dimensional column vectors δ and p are called the reduction and penalty vectors, respectively. Typically these vectors depend on load fees, the holding period, the initial portfolio wealth, and current and future portfolio contributions and withdrawals. Further, the elements of these vectors are non-negative. In one embodiment of block 615, the elements of these vectors are computed using the formulas for front-end loads given by equations (12). Note that because of the penalty term, the objective function is no longer a quadratic function of the target mix x, and also depends on xo, the N dimensional column vector of initial mix values.
b=∥x−x o∥
s=∥x o −x∥ (EQ #16)
In equation (16), the “buy” variable b represents the portion of the target mix that is purchased in addition to any current holdings. Similarly, the “sell” variable s represents the portion of the target mix that is sold from any current holdings. The following three properties of the vectors b and s are consequences of equations (16) and the maximum and minimum bounds on the vector x:
0≦b≦∥u−x o∥
0≦s≦∥x o−1∥
bTs=0 (EQ #17)
The first two properties are simple bounds on the feasible values of b and s. Note that the lower bounds restrict the elements of b and s to non-negative values. The third property is a complementarity condition. Together with the non-negativity constraints, the complementarity condition requires that at least one of the ith elements of b and s be zero. The financial implication of this constraint is to prevent a simultaneous sale and purchase of the same position in a portfolio. A last property of equations (16) is a formula for the original vector x of decision variables in terms of xo, b, and s:
x=x o +b−s (EQ #18)
Using equation (18) and a feasible pair of buy-sell vectors, a feasible value of the target mix vector can be constructed. In particular, if b* and s* are the optimal feasible vectors for a program in b and s, then the optimal portfolio mix x* can be computed.
The ith element of the vector min[x,y] (max[x,y]) is equal to the minimum (maximum) xvalue of x1 and yi, the ith elements of the vectors x and y. Note that v represents the amount of the target mix x that is both above the lower bound and is immune to the load penalty term, and that w represents the amount of the target mix x that is above the lower bound and is subject to the penalty term. The following three properties of the vectors v and w are consequences of equations (16) and the maximum and minimum bounds on the vector x:
0≦v≦min[u,x o]−min[l,x o]
0≦w≦max[u,x o]−max[l,x o]
(v−∥x o −l∥)T(w+∥l−x o∥)=0 (EQ #20)
The first two properties are simple bounds on the feasible values v and w. Note that the lower bounds restrict the elements of v and w to non-negative values. The third property is a complementarity condition. Again, the financial implication of this constraint is to prevent a simultaneous sale and purchase of the same position in a portfolio. A last property of equations (20) is a formula for the original vector of decision variables x in terms of l, v, and w:
x=l+v+w (EQ #21)
Using equation (21) and a feasible pair of immune-penalty vectors, a feasible value of the target mix vector can be constructed. In particular, if v* and w* are the optimal feasible vectors for a program in v and w, then the optimal portfolio mix x* can be computed.
Note that the objective function is a quadratic function of the 2N decision variables given by the elements of the vectors v and w. The term Φo is a constant with respect to the optimization, and can be ignored. Similarly, x can be eliminated from the multiple-account budget constraints given by equations (14):
It is further noted that if the bounds on v and w given by equations (22) are satisfied, then the bounds on x are automatically satisfied. Again, it can be shown that since the penalty vector p is non-negative, that the complementarity condition does not need to be explicitly enforced. Therefore, equations (22) and (23), together with the bounds from equation (20) comprise a quadratic program in 2N decision variables with simple bounds and M linear equality constraint.
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EP1259896A1 (en) | 2002-11-27 |
CA2401073A1 (en) | 2001-08-30 |
AU2001238660B2 (en) | 2007-05-31 |
WO2001063458A2 (en) | 2001-08-30 |
US20020002521A1 (en) | 2002-01-03 |
AU3866001A (en) | 2001-09-03 |
CN1449528A (en) | 2003-10-15 |
JP2004502210A (en) | 2004-01-22 |
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